Displaying similar documents to “On stabbing triangles by lines in 3-space”

On the structural result on normal plane maps

Tomás Madaras, Andrea Marcinová (2002)

Discussiones Mathematicae Graph Theory

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We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by 6 + [ ( 2 D + 12 ) / ( D - 2 ) ] ( ( D - 1 ) ( t - 1 ) - 1 ) . This improves a recent bound 6 + [ ( 3 D + 3 ) / ( D - 2 ) ] ( ( D - 1 ) t - 1 - 1 ) , D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.

On the number of intersections of two polygons

Jakub Černý, Jan Kára, Daniel Král', Pavel Podbrdský, Miroslava Sotáková, Robert Šámal (2003)

Commentationes Mathematicae Universitatis Carolinae

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We study the maximum possible number f ( k , l ) of intersections of the boundaries of a simple k -gon with a simple l -gon in the plane for k , l 3 . To determine the number f ( k , l ) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k l ) the easy upper bound k l - l to k l - k / 6 - l and obtain exact bounds for k = 5 ( f ( 5 , l ) = 4 l - 2 ) in this case.

How many clouds cover the plane?

James H. Schmerl (2003)

Fundamenta Mathematicae

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The plane can be covered by n + 2 clouds iff 2 .

Sets invariant under projections onto two dimensional subspaces

Simon Fitzpatrick, Bruce Calvert (1991)

Commentationes Mathematicae Universitatis Carolinae

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The Blaschke–Kakutani result characterizes inner product spaces E , among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace F there is a norm 1 linear projection onto F . In this paper, we determine which closed neighborhoods B of zero in a real locally convex space E of dimension at least 3 have the property that for every 2 dimensional subspace F there is a continuous linear projection P onto F with P ( B ) B .

A method for evaluating the fractal dimension in the plane, using coverings with crosses

Claude Tricot (2002)

Fundamenta Mathematicae

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Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set E ε , we consider a new method based upon coverings of E ε with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula...

Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov (2007)

Colloquium Mathematicae

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Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by R B ( T ) the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science R B ( T ) is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein)....

On the f - and h -triangle of the barycentric subdivision of a simplicial complex

Sarfraz Ahmad (2013)

Czechoslovak Mathematical Journal

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For a simplicial complex Δ we study the behavior of its f - and h -triangle under the action of barycentric subdivision. In particular we describe the f - and h -triangle of its barycentric subdivision sd ( Δ ) . The same has been done for f - and h -vector of sd ( Δ ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h -triangle of Δ are nonnegative, then the entries of the h -triangle of sd ( Δ ) are also nonnegative. We conclude with a few properties of the h -triangle of sd ( Δ ) . ...

An axiomatics of non-Desarguean geometry based on the half-plane as the primitive notion

A. Śniatycki

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CONTENTSIntroduction................................................................................................................................................. 5PART I1. Axioms of Boolean algebra................................................................................................................. 62. Half-planes and their axioms.............................................................................................................. 73. The line.......................................................................................................................................................